AutoSamplingTheory.SALD.cycle69MainSkeletonAnalyticMiddleContract

Data definition / provenance and workflow record

Meaning and type

The result has data type AutoSamplingTheory.SALD.MainSkeletonAnalyticInterfaceLedger. A value of this type stores descriptions; it is not a proof of the statements in those descriptions.

def cycle69MainSkeletonAnalyticMiddleContract :
    MainSkeletonAnalyticInterfaceLedger

Construction and field-by-field explanation

Construct a data record from explicit fields and the audited defaults shown below.

This Lean definition constructs provenance or workflow data. It does not prove the mathematical statements stored as text. Status labels, named dependencies and citations are data, not compilation, proof or source certificates.

sourceBlock : AutoSamplingTheory.SourceAnchor (explicit)

Nested provenance data.

AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteDerivativeSource — audited data reference, not expanded and not a compiled dependency edge

objective : String (explicit)

Descriptive text even when field names say formula, theorem, inequality, derivative, source gap, or proof.

Cycle 69 middle: synchronize the upper post-route analytic-interface ledger against the full SALD theorem route, verify that the five slow analytic interfaces remain the named consumers for the six faithful theorem skeletons in paper order, and keep the next lower packet exactly on SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation / SALD.generalMovingTargetDiscreteDerivativeSideConditionContract / sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387.

sourceLabels : List String (explicit)

Ordered descriptive/naming checklist, not logical conjunction or compiler dependency list.

Ordered data items

1. lem:gronwall
2. lem:dv_variation
3. eq:LSI-KL-FI
4. proof:thm:forward-KL:derivative
5. proof:thm:forward-KL-discrete:conditional-fp
6. proof:thm:general-moving-target-SALD:derivative
7. proof:thm:general-moving-target-SALD-discrete:em-conditional-fp
8. thm:forward-KL
9. thm:forward-KL-discrete
10. prop:guided_path_residual
11. thm:general-moving-target-SALD
12. thm:unified-forward-KL
13. thm:general-moving-target-SALD-discrete

analyticInterfaces : List String (explicit)

Ordered descriptive/naming checklist, not logical conjunction or compiler dependency list.

Ordered data items

1. Middle five-backend check, Gronwall: theorem routes still consume SALD.saldGronwallEndpointCalculusContract and theorem-specific Gronwall side-condition obligations for endpoint-safe differentiability or right-derivative/FTC, interval integrability, endpoint evaluation, coefficient regularity, exponent splitting, and display matching; lem:gronwall remains obligation.
2. Middle five-backend check, DV: all velocity and residual-energy uses still pass through dvVariationalFormulaInterface saldDvVariationSource plus theorem-specific common-space, absolute-continuity, finite-KL/log-likelihood, selected-test measurability, positive-alpha, and finite-log-mgf witnesses; lem:dv_variation remains source-cited plus obligations.
3. Middle five-backend check, LSI/KL/FI: all LSI uses still pass through SALD.saldLsiKlFiDensityTestContract, SALD.lsiKlFiDensityTestObligation, and probability.lsi_to_kl_fi with Radon-Nikodym density, zero-set convention, admissible sqrt-density test or approximation, entropy identity, finite KL/FI, and Fisher chain-rule obligations explicit.
4. Middle five-backend check, continuous Fokker-Planck/KL derivative: thm:forward-KL and thm:general-moving-target-SALD still consume SALD.forwardKlDerivativeSideConditionContract and SALD.generalMovingTargetDerivativeCandidateContract through the cycle-65 and cycle-67 theorem-route audits; scalar wrappers do not close the analytic backend.
5. Middle five-backend check, EM interpolation Fokker-Planck: thm:forward-KL-discrete and thm:general-moving-target-SALD-discrete still consume SALD.discreteForwardKlEmInterpolationSideConditionContract, SALD.generalMovingTargetDiscreteDerivativeSideConditionContract, cycle-63 endpoint laws, cycle-64 conditional-drift algebra, and sald.general_moving_target_discrete.em_interpolation_fp; conditional laws, density/AC, weak FP, KL differentiation, and stitched intervals remain obligations.

theoremRoute : List String (explicit)

Ordered descriptive/naming checklist, not logical conjunction or compiler dependency list.

Ordered data items

1. 1. thm:forward-KL: appendix.tex:168-252 remains continuous KL derivative/Fokker-Planck -> LSI/KL/FI -> DV velocity-energy -> Gronwall endpoint/exponent display.
2. 2. thm:forward-KL-discrete: appendix.tex:260-592 remains EM endpoint/conditional-FP -> KL derivative/frozen defect/LSI -> DV velocity -> time-changed Gronwall -> accumulated-error display.
3. 3. prop:guided_path_residual: appendix.tex:619-704 remains guided normalizer differentiation, quotient/product calculus, divergence cancellation, centered residual identity, and mean-zero residual.
4. 4. thm:general-moving-target-SALD: appendix.tex:724-949 remains continuous general KL derivative -> residual Young/LSI -> residual DV -> sigma-weighted Gronwall -> pure contraction.
5. 5. thm:unified-forward-KL: main_body.tex:359-395 and appendix.tex:949-951 remain the correction-field transport bridge and c_t=u_t, m_t=w_t specialization of thm:general-moving-target-SALD.
6. 6. thm:general-moving-target-SALD-discrete: appendix.tex:1313-1603 remains general EM endpoint/conditional-FP -> frozen-delta -> KL derivative/LSI -> residual DV -> constant-schedule Gronwall/display stitching.

modeDiscipline : List String (explicit)

Ordered descriptive/naming checklist, not logical conjunction or compiler dependency list.

Ordered data items

1. faithfulPaper Phase 1: this middle audit is source-to-Lean synchronization only, using original main_body.tex, appendix.tex, and iteration_complexity.tex; sald_version_2.tex remains excluded.
2. No theorem statement, coefficient, alpha range, sigma factor, endpoint law, source label, or proof order is changed.
3. All theorem contracts remain ProofStatus.contractOnly and all slow analytic interfaces remain ProofStatus.obligation or ProofStatus.sourceCited unless their full analytic dependencies compile locally.
4. The only lower packet kept active by this middle pass is the shared EM interpolation conditional-law/Fokker-Planck backend over appendix.tex:1358-1387.

nonGoals : List String (explicit)

Ordered descriptive/naming checklist, not logical conjunction or compiler dependency list.

Ordered data items

1. Do not start broad SLT, disintegration, SDE, Gaussian concentration, entropy-duality, or reusable API backfill.
2. Do not turn cycle-63 endpoint-law bookkeeping or cycle-64 conditional-drift algebra into a proof of regular conditional laws, density/absolute-continuity, weak Fokker-Planck, or KL differentiation.
3. Do not replace the paper derivative -> LSI -> DV -> Gronwall route, the correction-field specialization, or the EM/frozen-delta route.
4. Do not mark Gronwall, DV, LSI/KL/FI, continuous Fokker-Planck/KL differentiation, EM conditional-FP, theorem contracts, or SLT reuse as formalized.

lowerPacket : List String (explicit)

Ordered descriptive/naming checklist, not logical conjunction or compiler dependency list.

Ordered data items

1. Target exactly SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation / SALD.generalMovingTargetDiscreteDerivativeSideConditionContract / sald.general_moving_target_discrete.em_interpolation_fp.
2. Source slice appendix.tex:1358-1366: keep common probability-space data, endpoint laws for hat rho_s and tilde pi_s, density/absolute-continuity, finite KL, mass conservation, and endpoint-safe KL differentiation prerequisites explicit.
3. Source slice appendix.tex:1368-1377: keep the regular conditional law of X_k^eta given hat X_s=x, the measurable/integrable conditional drift bar b_{k,s}(x), and conditional-expectation linearity as obligations; reuse cycle-64 algebra only after those inputs are supplied.
4. Source slice appendix.tex:1379-1387: state the weak conditional Fokker-Planck equation with exact source signs -div(hat rho_s bar b_{k,s}) and +(sigma_eta^2/2) Delta hat rho_s.
5. Keep KL differentiation after the weak FP identity, LSI/KL/FI, DV, frozen-delta, and Gronwall/display matching as separate downstream obligations.

reviewerChecklist : List String (explicit)

Ordered descriptive/naming checklist, not logical conjunction or compiler dependency list.

Ordered data items

1. SALD.cycle69MainSkeletonAnalyticMiddleObligation is listed by all six theorem contracts while those contracts remain ProofStatus.contractOnly.
2. SALD.cycle69MainSkeletonAnalyticInterfaceDag contains ASTIS.SALD.cycle69.middle_interface_audit before ASTIS.SALD.cycle69.lower_packet.em_interpolation_fp.
3. SALD.saldDependenciesForLabel includes SALD.cycle69MainSkeletonAnalyticMiddleContract, SALD.cycle69MainSkeletonAnalyticMiddleObligation, and sald.main_skeleton.cycle69_middle_interface_audit through cycle69MainSkeletonDependencyNames.
4. The conversion window, proof-obligation ledger, and SLT reuse audit record this middle synchronization and keep the selected lower packet narrow.
5. No theorem statement, source constant, source label, source-file scope, theorem status, SLT reuse status, or analytic backend status changes.
6. python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass.

dependencies : List String (explicit)

Ordered descriptive/naming checklist, not logical conjunction or compiler dependency list.

Ordered data items

1. SALD.cycle69MainSkeletonAnalyticInterfaceLedger
2. SALD.cycle69MainSkeletonAnalyticInterfaceObligation
3. SALD.cycle68UnifiedDiscreteGeneralSkeletonMiddleObligation
4. SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeLowerObligation
5. SALD.cycle67GuidedGeneralSkeletonMiddleObligation
6. SALD.cycle66DiscreteForwardKlSkeletonMiddleObligation
7. SALD.cycle65ForwardKlSkeletonMiddleObligation
8. SALD.cycle64MainSkeletonAnalyticMiddleObligation
9. SALD.saldGronwallEndpointCalculusContract
10. dvVariationalFormulaInterface saldDvVariationSource
11. SALD.saldLsiKlFiDensityTestContract
12. SALD.forwardKlDerivativeSideConditionContract
13. SALD.generalMovingTargetDerivativeCandidateContract
14. SALD.discreteForwardKlEmInterpolationSideConditionContract
15. SALD.generalMovingTargetDiscreteDerivativeSideConditionContract
16. SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation
17. SALD.generalMovingTargetDiscreteConditionalDriftContract
18. SALD.generalMovingTargetDiscreteConditionalDriftLinearCombination
19. SALD.generalMovingTargetDiscreteConditionalDriftFieldOfLinearCombination
20. sald.general_moving_target_discrete.em_interpolation_fp
21. sald.discrete_forward_kl.em_interpolation_fp

status : AutoSamplingTheory.ProofStatus (explicit)

Stored workflow tag; honor the exact default but do not infer mathematical certification.

AutoSamplingTheory.ProofStatus.obligation — stored label only; no proof certification

def cycle69MainSkeletonAnalyticMiddleContract :
    MainSkeletonAnalyticInterfaceLedger where
  sourceBlock := saldGeneralMovingTargetDiscreteDerivativeSource
  objective := "Cycle 69 middle: synchronize the upper post-route analytic-interface ledger against the full SALD theorem route, verify that the five slow analytic interfaces remain the named consumers for the six faithful theorem skeletons in paper order, and keep the next lower packet exactly on SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation / SALD.generalMovingTargetDiscreteDerivativeSideConditionContract / sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387."
  sourceLabels := [
    "lem:gronwall",
    "lem:dv_variation",
    "eq:LSI-KL-FI",
    "proof:thm:forward-KL:derivative",
    "proof:thm:forward-KL-discrete:conditional-fp",
    "proof:thm:general-moving-target-SALD:derivative",
    "proof:thm:general-moving-target-SALD-discrete:em-conditional-fp",
    "thm:forward-KL",
    "thm:forward-KL-discrete",
    "prop:guided_path_residual",
    "thm:general-moving-target-SALD",
    "thm:unified-forward-KL",
    "thm:general-moving-target-SALD-discrete"
  ]
  analyticInterfaces := [
    "Middle five-backend check, Gronwall: theorem routes still consume SALD.saldGronwallEndpointCalculusContract and theorem-specific Gronwall side-condition obligations for endpoint-safe differentiability or right-derivative/FTC, interval integrability, endpoint evaluation, coefficient regularity, exponent splitting, and display matching; lem:gronwall remains obligation.",
    "Middle five-backend check, DV: all velocity and residual-energy uses still pass through dvVariationalFormulaInterface saldDvVariationSource plus theorem-specific common-space, absolute-continuity, finite-KL/log-likelihood, selected-test measurability, positive-alpha, and finite-log-mgf witnesses; lem:dv_variation remains source-cited plus obligations.",
    "Middle five-backend check, LSI/KL/FI: all LSI uses still pass through SALD.saldLsiKlFiDensityTestContract, SALD.lsiKlFiDensityTestObligation, and probability.lsi_to_kl_fi with Radon-Nikodym density, zero-set convention, admissible sqrt-density test or approximation, entropy identity, finite KL/FI, and Fisher chain-rule obligations explicit.",
    "Middle five-backend check, continuous Fokker-Planck/KL derivative: thm:forward-KL and thm:general-moving-target-SALD still consume SALD.forwardKlDerivativeSideConditionContract and SALD.generalMovingTargetDerivativeCandidateContract through the cycle-65 and cycle-67 theorem-route audits; scalar wrappers do not close the analytic backend.",
    "Middle five-backend check, EM interpolation Fokker-Planck: thm:forward-KL-discrete and thm:general-moving-target-SALD-discrete still consume SALD.discreteForwardKlEmInterpolationSideConditionContract, SALD.generalMovingTargetDiscreteDerivativeSideConditionContract, cycle-63 endpoint laws, cycle-64 conditional-drift algebra, and sald.general_moving_target_discrete.em_interpolation_fp; conditional laws, density/AC, weak FP, KL differentiation, and stitched intervals remain obligations."
  ]
  theoremRoute := [
    "1. thm:forward-KL: appendix.tex:168-252 remains continuous KL derivative/Fokker-Planck -> LSI/KL/FI -> DV velocity-energy -> Gronwall endpoint/exponent display.",
    "2. thm:forward-KL-discrete: appendix.tex:260-592 remains EM endpoint/conditional-FP -> KL derivative/frozen defect/LSI -> DV velocity -> time-changed Gronwall -> accumulated-error display.",
    "3. prop:guided_path_residual: appendix.tex:619-704 remains guided normalizer differentiation, quotient/product calculus, divergence cancellation, centered residual identity, and mean-zero residual.",
    "4. thm:general-moving-target-SALD: appendix.tex:724-949 remains continuous general KL derivative -> residual Young/LSI -> residual DV -> sigma-weighted Gronwall -> pure contraction.",
    "5. thm:unified-forward-KL: main_body.tex:359-395 and appendix.tex:949-951 remain the correction-field transport bridge and c_t=u_t, m_t=w_t specialization of thm:general-moving-target-SALD.",
    "6. thm:general-moving-target-SALD-discrete: appendix.tex:1313-1603 remains general EM endpoint/conditional-FP -> frozen-delta -> KL derivative/LSI -> residual DV -> constant-schedule Gronwall/display stitching."
  ]
  modeDiscipline := [
    "faithfulPaper Phase 1: this middle audit is source-to-Lean synchronization only, using original main_body.tex, appendix.tex, and iteration_complexity.tex; sald_version_2.tex remains excluded.",
    "No theorem statement, coefficient, alpha range, sigma factor, endpoint law, source label, or proof order is changed.",
    "All theorem contracts remain ProofStatus.contractOnly and all slow analytic interfaces remain ProofStatus.obligation or ProofStatus.sourceCited unless their full analytic dependencies compile locally.",
    "The only lower packet kept active by this middle pass is the shared EM interpolation conditional-law/Fokker-Planck backend over appendix.tex:1358-1387."
  ]
  nonGoals := [
    "Do not start broad SLT, disintegration, SDE, Gaussian concentration, entropy-duality, or reusable API backfill.",
    "Do not turn cycle-63 endpoint-law bookkeeping or cycle-64 conditional-drift algebra into a proof of regular conditional laws, density/absolute-continuity, weak Fokker-Planck, or KL differentiation.",
    "Do not replace the paper derivative -> LSI -> DV -> Gronwall route, the correction-field specialization, or the EM/frozen-delta route.",
    "Do not mark Gronwall, DV, LSI/KL/FI, continuous Fokker-Planck/KL differentiation, EM conditional-FP, theorem contracts, or SLT reuse as formalized."
  ]
  lowerPacket := [
    "Target exactly SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation / SALD.generalMovingTargetDiscreteDerivativeSideConditionContract / sald.general_moving_target_discrete.em_interpolation_fp.",
    "Source slice appendix.tex:1358-1366: keep common probability-space data, endpoint laws for hat rho_s and tilde pi_s, density/absolute-continuity, finite KL, mass conservation, and endpoint-safe KL differentiation prerequisites explicit.",
    "Source slice appendix.tex:1368-1377: keep the regular conditional law of X_k^eta given hat X_s=x, the measurable/integrable conditional drift bar b_{k,s}(x), and conditional-expectation linearity as obligations; reuse cycle-64 algebra only after those inputs are supplied.",
    "Source slice appendix.tex:1379-1387: state the weak conditional Fokker-Planck equation with exact source signs -div(hat rho_s bar b_{k,s}) and +(sigma_eta^2/2) Delta hat rho_s.",
    "Keep KL differentiation after the weak FP identity, LSI/KL/FI, DV, frozen-delta, and Gronwall/display matching as separate downstream obligations."
  ]
  reviewerChecklist := [
    "SALD.cycle69MainSkeletonAnalyticMiddleObligation is listed by all six theorem contracts while those contracts remain ProofStatus.contractOnly.",
    "SALD.cycle69MainSkeletonAnalyticInterfaceDag contains ASTIS.SALD.cycle69.middle_interface_audit before ASTIS.SALD.cycle69.lower_packet.em_interpolation_fp.",
    "SALD.saldDependenciesForLabel includes SALD.cycle69MainSkeletonAnalyticMiddleContract, SALD.cycle69MainSkeletonAnalyticMiddleObligation, and sald.main_skeleton.cycle69_middle_interface_audit through cycle69MainSkeletonDependencyNames.",
    "The conversion window, proof-obligation ledger, and SLT reuse audit record this middle synchronization and keep the selected lower packet narrow.",
    "No theorem statement, source constant, source label, source-file scope, theorem status, SLT reuse status, or analytic backend status changes.",
    "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass."
  ]
  dependencies := [
    "SALD.cycle69MainSkeletonAnalyticInterfaceLedger",
    "SALD.cycle69MainSkeletonAnalyticInterfaceObligation",
    "SALD.cycle68UnifiedDiscreteGeneralSkeletonMiddleObligation",
    "SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeLowerObligation",
    "SALD.cycle67GuidedGeneralSkeletonMiddleObligation",
    "SALD.cycle66DiscreteForwardKlSkeletonMiddleObligation",
    "SALD.cycle65ForwardKlSkeletonMiddleObligation",
    "SALD.cycle64MainSkeletonAnalyticMiddleObligation",
    "SALD.saldGronwallEndpointCalculusContract",
    "dvVariationalFormulaInterface saldDvVariationSource",
    "SALD.saldLsiKlFiDensityTestContract",
    "SALD.forwardKlDerivativeSideConditionContract",
    "SALD.generalMovingTargetDerivativeCandidateContract",
    "SALD.discreteForwardKlEmInterpolationSideConditionContract",
    "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract",
    "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation",
    "SALD.generalMovingTargetDiscreteConditionalDriftContract",
    "SALD.generalMovingTargetDiscreteConditionalDriftLinearCombination",
    "SALD.generalMovingTargetDiscreteConditionalDriftFieldOfLinearCombination",
    "sald.general_moving_target_discrete.em_interpolation_fp",
    "sald.discrete_forward_kl.em_interpolation_fp"
  ]
  status := ProofStatus.obligation

/-- Cycle-69 middle obligation tying the post-route analytic-interface audit
to the selected shared EM conditional-law/Fokker--Planck backend. -/